Applied Mathematics and Nonlinear Sciences

The author analyzes the network marketing model under the e-commerce mode through the marketing function under the nonlinear differential equation, through the research, it is found that the four variables of commodity price, advertising investment, sales model design investment and logistics investment are the key explanatory variables of sales volume of marketing network, among them, the price of goods has the greatest impact on online marketing, secondly, advertising investment, logistics investment and design investment have less impact


Introduction
With the continuous development of network technology and Internet sales, e-commerce has developed rapidly.Under this great development trend, enterprises have become an important means of sales by using the Internet for product marketing.Using mathematical thinking to scientifically select and develop foot marketing under e-commerce mode plays an important role in the development of e-commerce [1].
The network marketing under the e-commerce mode is an important marketing activity, and an important means for enterprises to develop their own marketing channels and channels, and constantly explore new influence directions (Figure 1).The use of online marketing model for official product promotion and promotion is important for the development of enterprise marketing scale, it is of great significance to further expand the sales of enterprise products.In online marketing, we should apply mathematical thinking to research and analyze the necessary marketing strategies and methods, and then provide a strong theoretical and mechanism guarantee [2].First of all, mathematical thinking is a rational way of thinking.It is an important tool for network marketing analysis and strategy research.Secondly, the research and analysis of network marketing strategies and methods by using mathematical thinking, as well as the feasibility study of formulating necessary network marketing strategies by using mathematical thinking, are the important foundation to ensure the smooth implementation of network marketing and the success of network marketing, and also the important method for enterprises to promote network marketing mode.The penetration of mathematical thinking into online marketing itself is a promotion and promotion of online marketing, and an important theoretical guarantee to ensure the effective integration and operation of the overall marketing mode of enterprises [3].

Marketing Function in Nonlinear Differential Equation
The profit of marketing is mainly determined by the difference between sales revenue and sales cost, in which sales revenue depends on the product of the net price of goods and their sales volume; Marketing cost is mainly divided into variable marketing cost, fixed cost, marketing and other costs, the profit function of marketing can be expressed as follows: (1) Among them,  represents the total profit in marketing; P represents the original price of the market sale; Q refers to the quantity of goods in marketing; K represents the discount amount in commodity marketing;  !represents the variable cost of each commodity;  " represents the fixed cost of goods in the market;  # represents the marketing and other costs of goods in the market.
It can be concluded from Formula (1) that it is necessary to produce and sell the commodity only when it is ( − ) ≥  !, that is, the selling price of the commodity in the market is not less than the variable production cost.If it is ( − ) <  !, it indicates that the sales of the goods are loss making and the fixed cost and variable cost of the goods cannot be compensated.
In order to analyze the profit model in marketing in more detail, the author needs to use the sales equation that affects the sales volume to study the reaction of commodity sales in the market [4].Assuming that the sales volume is mainly determined by two types of variables, namely, controllable factors and uncontrollable factors, the sales function in marketing can be expressed as follows: (2) Among them, ( $ ,  % , . . .,  # ) represents controllable sales factors in marketing; ( $ ,  % , . . .,  & ) refers to uncontrollable sales factors in marketing.
Variable X includes such factors as the price of goods in the market, discount for promotion, variable cost of goods and marketing cost; The variable Y includes market size, competitors' prices, consumer income and other factors, when these uncontrollable factors change, the purchase rate of the market will also change.Although the enterprise cannot control the influence of variable Y on sales volume, it can predict and estimate according to relevant data and data.We assume that the uncontrollable market sales variable Y has been completely estimated and predicted, then the sales function of formula ( 2) can be rewritten as follows: ( The above formula represents the function that the sales volume is affected by the controllable variable X under the given uncontrollable influencing factor conditions in the market sales.Then, the sales function can be expressed as follows: (4) Where Q represents the sales volume in marketing; p represents the original price of the market sale; K represents the discount amount in the commodity promotion;  !represents the variable cost of goods;  # refers to the marketing cost of goods in the market.
Since the marketing cost of goods mainly includes advertising, salaries of salesmen, research and design of sales methods and other factors, formula (4) can be rewritten as follows: (5) Where, A represents the input cost of advertising in the promotion, W represents the additional salary of promoters in the promotion, and R&D represents the cost of research and design of sales methods; (P,k,  !, A, W,R&D) are the key marketing factors of the marketing mix [5][6].
How to transform formula (5) into the optimal marketing mix is the core of the limited allocation of marketing resources.Since then, we have used Cobb Douglas' production function to reflect the sales response function under the limited allocation of marketing resources, which can be expressed as follows: ( ) Where Q represents the sales volume in marketing; P represents the original price of the market sale; A represents the input cost of advertising in promotion; R represents the cost of research and design of sales methods; k represents the discount amount in the commodity promotion; b, c and d are constants and sales elasticity of each factor.Take the logarithm on both sides of equation ( 6) to obtain the linear function of sales volume on controllable influencing factors, namely: (7) Among them, the meaning of each parameter variable is the same as that in equation ( 6), which is just for parameter estimation and equation simplification.

Application of logical thinking in online marketing
Logical thinking is one of the basic thinking modes of mathematical thinking, its rich and important feature is that everything should develop and progress according to certain logical reasoning.Under the e-commerce mode, network marketing should also follow this thinking mode.According to the viewpoint of logical thinking, there is an inevitable and regular relationship between the seller and the consumer, which is stable and relatively changeable.It is of great significance for the formulation and selection of network marketing strategies to find this law according to the development of logical thinking and make use of it.First of all, under the logical thinking mode, the buyers and sellers of online marketing are psychologically in a relationship of checks and balances [7][8].That is to say, in online marketing, the relationship between the buyer and the seller is mutually connected, not a unilateral supply or demand relationship.Therefore, in order to maximize benefits, we must achieve a win-win situation.Therefore, when choosing the online marketing model, we should try to choose a win-win marketing strategy, we should not simply consider the interests of one party.Secondly, logical thinking requires that we should pay attention to the reasoning of the buying and selling process in online marketing, that is, whether it conforms to objective laws and social norms.Under the condition of logical thinking, online marketing should follow the pattern and rules set by the week and be carried out under certain rules [9].Therefore, there are certain laws in online marketing.Therefore, in online marketing, we should pay attention to grasp the reasoning of the buying and selling process and conform to certain laws.Logical thinking is widely used in online marketing, it is used to analyze the needs of both parties and formulate relevant marketing actions; Making marketing strategies according to objective laws and actual conditions is the inevitable choice of network marketing under the modern e-commerce model, and the only way to healthy development [10][11][12].

Application of divergent thinking in online marketing
Open thinking refers to the thinking mode that conducts multi aspect and all-round thinking operation within a certain range according to certain standards and requirements, and then seeks the optimal solution.Divergent thinking requires scattered but not chaotic thinking, that is, decentralized thinking, multi-directional and multi angle consideration, but not aimless thinking.Divergent thinking is of great significance and has been widely used in e-commerce.Internet marketing methods vary greatly.
The market environment is changeable, controlling such a market environment, relying solely on traditional thinking mode and adopting a single marketing strategy is not only difficult to achieve the marketing purpose, but will have a negative impact on the image of the enterprise and the sales of chrome body [13][14].Use open thinking to think and understand the changing environment of online marketing from multiple perspectives, and then deeply study the market situation, accurately grasp the market dynamics, so as to maximize the benefits and effects of online marketing.First of all, divergent thinking requires network marketing personnel to have strong creativity and innovation ability.Divergent thinking requires network marketers to constantly change their own coping strategies and influence methods according to the market situation, and use the most effective way to carry out marketing activities.The network marketing is different from the real marketing model, and its key feature is that the network marketing is significantly higher than the marketing in the real environment in terms of communication speed and environmental complexity.Therefore, it is also difficult to control the market.The use of divergent thinking in online marketing requires online marketers to enhance their creativity and innovation to adapt to the speed and complexity of this development.
Secondly, divergent thinking requires online marketers to constantly update their ideas.Only when we fully understand and control the market situation of network marketing can we fundamentally play the role of network marketing, realize the maximum development of network marketing, and further enhance the role of network marketing in the enterprise marketing system and enhance the role of network marketing [15].Thirdly, the use of divergent thinking to conduct market research, especially online market research is an important guarantee for effective online marketing.Network is a special market environment with unique characteristics, which determine the difference between network marketing and real marketing.Divergent thinking is widely used in online marketing, not only for research on marketing strategies and marketing methods, but also more importantly, in the process of online marketing, marketing personnel should constantly improve their awareness of the online market and control over online consumers, so as to improve their marketing skills and promotion capabilities.Using divergent thinking to plan online marketing activities and research and implement online promotion is an important means of online marketing, and divergent thinking will play a more extensive role in online marketing [16][17].

Application analysis of marketing function under e-commerce mode
The online marketing function model under the e-commerce mode is basically the same as the marketing function in the physical market, the difference is that the online marketing advertisement of e-commerce mainly depends on the website to publish, and the cost of network maintenance and information security needs to be added, which saves the cost of the physical store.In addition, an important controllable variable needs to be provided under the e-commerce model, which is the logistics variable.At this point, we can rewrite formula (8) as follows: Where, L represents the logistics cost under the e-commerce mode; E represents the constant in the logistics parameters to be estimated; The meanings of other variables are the same as those in equation ( 8) [18][19].Suppose Company A is selected as an example, experts are used to estimate the sales volume of each marketing mix, and the variable data of estimated sales volume, commodity price, advertising cost, design maintenance cost, logistics cost and other factors are substituted into Formula (9) for OLS estimation.The following parameter estimation model can be obtained: From the regression results of company A's marketing model, it can be seen that the p of the four explanatory variables are all less than 0.05, indicating that they are significant underwater at 5% significance.The absolute coefficient of the model is 0.95, indicating that these four explanatory variables can explain 95% of the proposed sales volume of goods, which means that the selection of model variables is more reasonable.The price of goods is inversely related to the sales volume, every 1% decrease in the price will increase the sales volume by 2.38 percentage points.Similarly, for every 1% increase in advertising investment in e-commerce, the online marketing volume will increase by 0.16 percentage points; Each 1% increase in the design input of the sales model and scheme will increase the sales volume by 0.07%, and each 1% increase in the logistics cost will increase the online marketing volume by 0.02% [20].

Conclusion
To sum up, nonlinear differential equation is an important tool to quantify and optimize online marketing methods under the e-commerce model, with the continuous development of e-commerce and the in-depth complexity of online marketing, the application advantages of nonlinear differential equation and mathematical ideas will become increasingly prominent.Therefore, the method of nonlinear differential equation is adopted to study the online marketing model under the e-commerce model, it can not only improve the sales volume and profit income in e-commerce, but also optimize the internal network marketing mode of e-commerce.